1 Ancient Egypt
Summary
• Recorded civilization in Nile valley from c. 3150 BC.
• Conquered 322 BC by Alexander the Great (Greece/Macedonia).
• Became a Roman province in 30 BC under Cleopatra.
• Discovery of the Rosetta Stone (AD 1799) containing Greek, hieroglyphic and demotic (post
hieratic, cursive) Egyptian script, allowed translation of ancient Egyptian writing.
• Primary mathematical sources: Rhind/Ahmes (A’h-mose) papyrus c. 1650 BC and the Moscow
papyrus c. 1700 BC.
1
Part of the Rhind papyrus is shown below. It contained two tables: unit
fraction representations of all
2
n
for n < 100 (ish) and expressions for
3
10
,
4
10
, . . . ,
9
10
in terms
of unit fractions. Also included were around 100 worked problems. Scribes would learn the
method by copying previous problems and changing the numbers.
The Moscow papyrus is shorter and more focused on geometric problems.
• Few other primary sources. Egyptians wrote on papyrus (plant-based form of paper) which
decomposes. Other Egyptian mathematics was likely absorbed uncredited by the Greeks.
• Practical/non-theoretical: worked problems on sums, linear equations, construction and land-
measurement (tax-collection). No clear distinction between exact and approximate solutions.
1
Rhind was a Scottish egyptologist. Ahmes was the name of the scribe who wrote/copied the papyrus. The Moscow
papyrus is named because it was sold to the Moscow Museum of Fine Art; its author is unknown.
1
Notation & Egyptian Fractions
The ancient Egyptians had two distinct systems for enumeration: hieroglyphic (dating at least to
5000 BC) and hieratic (c. 2000 BC). These changed over time, so we give only one version.
Hierogylphic enumeration Essentially decimal symbols for
numerals/powers of 10.
• Could be written in any direction: top-to-bottom, right-to-
left, etc., or just lumped together: e.g.
2349 =
• Slow to write, numbers take up a lot of space.
Numeral Hieroglyph
1
10 (heel bone)
100 (snare)
1000 (lotus flower)
10000 (finger)
100000 (fish)
1000000 (person)
Hieratic enumeration We will largely ignore this
since it is written cursively.
• Different symbols for 1–9, 10–90, 100–900
etc., mapped onto hieratic alphabet.
• System copied later by the Greeks with their
own alphabet.
• Pros: less space, easier to write with ink,
each number requires fewer symbols.
• Cons: More symbols, slower calculations.
For instance, 23 would be written (approximately!) |||∧.
The Eygptians had no numeral for zero, though the hieroglyph nfr (beautiful/perfect) was used to
denote, for instance, the base floor of a building or to indicate balanced books in accounting.
Fractions Ancient Egyptians worked almost entirely with reciprocals of integers (
1
n
where n ∈ N).
In modern times, any fraction represented as a sum of reciprocals is called an Egyptian fraction; their
theory is still actively researched. For instance
1
2
+
1
4
+
1
5
is a representation of
19
20
as an Egyptian fraction.
In hieratic notation, a dot was placed above to denote the reciprocal: e.g.,
˙
∧ is
1
10
.
In hieroglyphs, a reciprocal was represented by plac-
ing an oval over a numeral. We do this with a bar:
e.g. 2 =
1
2
. As with integers, combinations of frac-
tions could be written in any order/direction.
The only non-reciprocal fractions with special sym-
bols were
2
3
and
3
4
, and these only appeared late in
Egyptian civilization.
2
Fraction Hieroglyph Modern
1
3
3
1
41
41
1
103204
103204
31 +
1
2
+
1
25
25 2 31
2
For instance an oval over one-and-a-half sticks for
2
3
, and an oval over three short-long-short sticks for
3
4
.
2
The Rhind papyrus contains a table, of which we reproduce part,
showing how to express
2
n
as Egyptian fractions for all odd integers
n < 100. The first column denotes n, and the remaining columns the
Egyptian fraction representation. For instance, the first row states
2
5
=
1
3
+
1
15
( 3 15)
5 3 15
7 4 28
9 6 18
11 6 66
13 8 52 104
15 10 30
There are several approaches to finding Egyptian fraction representations, and it can be proved that
any fraction may be written in such a form. If the denominator is odd, then a simple place to start is
with
2
mn
=
1
mr
+
1
nr
where r =
m + n
2
Most lines in the Rhind table follow this formula, but not all. Note also that the formula often permits
multiple representations.
• The line for
2
5
has m = 1, n = 5 and r = 3; this is unique up to reordering.
•
2
9
may be represented
2
9
=
(
1
9
+
1
9
(m, n, r) = (3, 3, 3)
1
5
+
1
45
(m, n, r) = (1, 9, 5)
The first of these is essentially useless and the second is not the expression from the table.
• In the table,
2
13
is written as a sum of three reciprocals instead of two.
The table reduced the need to divide and sped up the computation of harder fractions: for instance,
5
13
= 2 ·
2
13
+
1
13
= 2
1
8
+
1
52
+
1
104
+
1
13
=
1
4
+
1
13
+
1
26
+
1
52
Egyptian Calculations & Example Problems
Addition/Subtraction With hieroglyphs this is simple: Write out numbers one above another and
count up the symbols! Replace 10 of one by the next symbol. For subtraction, one might need to
convert a larger symbol to 10 of a smaller one. Essentially this is ‘carrying.’ A special symbol was
used to denote both addition and subtraction: its meaning changed depending on the direction the
text was read.
Multiplication This relied on a base-two algorithm. To compute ab:
1. Write 1, b
2. Repeatedly double each until the first term is about to exceed a
3. Determine powers of 2 that sum to a
4. Sum the corresponding multiples of b
3
For example, to compute 13 · 15, we construct a table where the checked rows are summed.
1 15 ✓
2 30
4 60 ✓
8 120 ✓
=⇒ 13 · 15 = (1 + 4 + 8) · 15 = 15 + 60 + 120 = 195
Note how 13 = 1 + 4 + 8 = 2
0
+ 2
2
+ 2
3
is essentially the binary representation. We stopped at the
fourth row since another doubling would have resulted in the first term (16) exceeding 13. We could
instead have reversed the roles of the factors:
1 13 ✓
2 26 ✓
4 52 ✓
8 104 ✓
=⇒ 15 · 13 = (1 + 2 + 4 + 8) · 13 = 13 + 26 + 52 + 104 = 195
All you need is addition and the ability to multiply by 2!
Division This also relies on doubling/halving, though the answer is non-unique and might re-
quire some creativity. To find
a
b
, think about solving the problem a = bx and apply a variant of the
multiplication algorithm to find multiples of b summing to a. Here are some examples.
1. To compute 260 ÷ 13, we repeatedly double 13 until terms in the right column sum to 260.
1 13
2 26
4 52 ✓
8 104
16 208 ✓
Since 260 = 208 + 52 we conclude that 260 ÷ 13 = 16 + 4 = 20
2. To find 5 ÷ 13 we start by dividing by 2 with the intent of making terms in the right column
sum to 5.
1 13
2 6 2
4 3 4 ✓
8 1 2 8 ✓
Since (3 4) + (1 2 8) = 4 2 4 8 is 8 (
1
8
) short of what we want, we continue the table in a different
way. First divide by 13, then continue halving until we obtain the desired 8 in the right column.
13 1
26 2
52 4
104 8 ✓
We conclude that 5 ÷ 13 = 4 8 104 =
1
4
+
1
8
+
1
104
. We could have proceeded differently to obtain
the same result as followed from the Rhind table:
5 = (3 4) + 1 + 2 + 4 =⇒ 5 ÷ 13 = 4 13 26 52
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Practical application: Loaf-splitting A typical Egyptian problem might involve determining how
to split 5 loaves among 13 people. The previous calculation tells us that we could give each person
1
4
+
1
8
+
1
104
of a loaf. This might seem complicated but it has some advantages over the modern
approach where we’d first cut each loaf into 13 equal pieces:
• The large chunks of bread are created by repeatedly cutting in half: this is easy to do accurately,
whereas cutting 13
th
parts is difficult! The remaining 104
th
parts of a loaf would probably be
ignored as crumbs.
• The Egyptian approach only requires 34 cuts, as opposed to 60 in the modern style.
Linear equations Another common type of problem was how to solve linear equations. Solutions
were based on the method of false position. Essentially one guesses an approximate solution, then
modifies it until it works. Here is problem 24 of the Rhind papyrus.
A heap plus a seventh of a heap is nineteen. What is the heap?
In modern algebra, representing ‘heap’ by x, we wish to solve x +
1
7
x = 19. Here is the Egyptian
method.
1. Guess intelligently: x = 14 is easy to divide by 7 and we obtain
x +
1
7
x = 16
2. Correct our guess: We want 19, not 16, so we multiply our guess
(14) by
19
16
= 1 8 16 to obtain the correct answer
x = 2 4 8 + 4 2 4 + 9 2 = 16 2 8
1 1 8 16
2 2 4 8 ✓
4 4 2 4 ✓
8 9 2 ✓
Compare this with the ‘modern’ method:
x +
1
7
x = 19 =⇒
8
7
x = 19 =⇒ x =
7 · 19
8
=
133
8
=
128 + 5
8
= 16
5
8
Are the any benefits to the Egyptian approach?
Geometry Problem 48 in the Rhind papyrus involves using an
octagon to approximated the area of a circle. A square of side 9
is drawn, where each side is split into thirds and the four corner
squares are cut in half. The area of the octagon is then
81 − 18 = 63
Since the area of the circle is
81π
4
, this amounts to the approximation
π ≈
28
9
= 3.11111 . . .
Problem 50 compares the same circle to a square of side 8: essen-
tially
81π
4
≈ 64 which corresponds to π ≈
256
81
= 3.16049 . . .
No explanation is given as to what inspired these methods, nor whether the scribes understood that
these were only approximations.
5
Other problems computed/approximated areas and volumes of trian-
gles, quadrilaterals, boxes, cylinders, and truncated pyramids. Again,
these were worked examples without general formulæ. The Egyp-
tians even had a notion of cotangent which they called the seked, use-
ful for describing and calculating slopes.
a
h
o
θ
seked =
a
o
= cot θ
Exercises Most of these problems are taken from the ‘official’ textbook.
1. Use Egyptian techniques to multiply 34 by 18 and to divide 93 by 5.
2. Use Egyptian techniques to multiply 28 by 1 + 2 + 4 (problem 9 of the Rhind papyrus).
3. A part of the Rhind papyrus table for division by 2 reads as follows:
2 ÷ 11 = 6 + 66, 2 ÷ 13 = 8 + 52 + 104, 2 ÷ 23 = 12 + 276
The calculation of 2 ÷ 13 is given below, where the right hand side is a modern rendering, and
the terminal ‘2’ indicates that the last entry in the right-hand column is indeed 2:
1 13
2 6 2
4 3 4
8 1 2 8
52 4
104 8
8 52 104 1 2 4 8 8
2
Perform similar calculations for 2 ÷ 11 and 2 ÷ 23.
(If you want the exact results from the papyrus, you’ll need to work with
2
3
: denote this by 3)
4. Draw a picture of 5 loaves to help describe how the Egyptians might have divided them be-
tween seven people.
5. Use the method of false position to solve problem 28 of the Rhind papyrus:
A quantity and its 2/3 are added together, and from the sum 1/3 of the sum is sub-
tracted, and 10 remains. What is the quantity?
6. Calculate a quantity such that if it is taken two times along with the quantity itself, the sum
comes to 9 (problem 25 of the Moscow papyrus).
7. (a) Find all ways in which
2
13
=
1
a
+
1
b
can be written as a sum of reciprocals (a ≤ b ∈ N).
(b) Repeat your calculation for
2
p
whenever p is an odd prime.
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